The red particle is going round the
circular path in anti-clock wise sense (considered positive)
with a uniform speed and its x-projection and y- projection
are shown in cyan and orange respectively. If the angular
position of the red particle is denoted by then its x and y
co-ordiantes are R cos and R sin ( With x to the right and y upward and origin at
the center of the cirlcle). But since the motion of the
particle round the circle is uniform , where omega is the angular speed and is the intial angular
position of the red particle. In the applet delta is set to
zero. is shown
by the small blue filled arc at the the center. In the applet
the phase is reset to zero after every revolution. In
practice it would increase continuously, going up by 2PI
radians for every revolution.

The cyan particles motion can be
represented as

and that of orange particle as

where A is the amplitude of SHM which is
equal to the radius of the circle and is called the phase. is the initial phase.

You should be able to see that eliminating
time from x and y equations gives you the equation of the
circle. Please note that even if varies in a more complicated manner
this would be true. But the x and y component motions then
would not be simple harmonic.

Do notice how the velocity vectors of the
cyan and orange particles vary. You should be able to see
that the velocity vectors diminish more rapidly when the
particles are farther from the center, indicating greater
acceleration when farther from the center of the circle.
Recall that in SHM acceleration is proportional to the
negative of displacement. That negative sign comes in because
when displacement is positive and increasing, the velocity
vector would be positive and decreasing. and if displacement
is positive and decreasing, the velocity vector would be
negative and increasing. You can check the cases when
displacement is negative.